a(1+cosC)+c(1+cosA)=a+(a²+b²-c²)/(2b)+c+(b²+c²-a²)/(2b)=a+c+(2b²)/(2b)=a+b+c=3b
a+c=2b
a,b,c成等差数列
设公差为d
cosB=[(b-d)²+(b+d)²-b²]/[2(b-d)(b+d]=(b²+2d²)/(2b²-2d²)=3b²/(2b²-2d²)-1≥3b²/2b²-1=3/2-1=1/2
当d=0时取等号